Foraging Model

1 Foraging Model

The proportion of time an animal is in a feeding behavioral state.

Process Model

\[Y_{i,t+1} \sim Multivariate Normal(d_{i,t},σ)\]

\[d_{i,t}= Y_{i,t} + γ_{s_{i,g,t}}*T_{i,g,t}*( Y_{i,g,t}- Y_{i,g,t-1} )\]

\[ \begin{matrix} \alpha_{i,1,1} & \beta_{i,1,1} & 1-(\alpha_{i,1,1} + \beta_{i,1,1}) \\ \alpha_{i,2,1} & \beta_{i,2,1} & 1-(\alpha_{i,2,1} + \beta_{i,2,1}) \\ \alpha_{i,3,1} & \beta_{i,3,1} & 1-(\alpha_{i,3,1} + \beta_{i,3,1}) \\ \end{matrix} \] \[logit(\phi_{Behavior}) = \alpha_{Behavior_{t-1}} \] The behavior at time t of individual i on track g is a discrete draw. \[S_{i,g,t} \sim Cat(\phi_{traveling},\phi_{foraging},\phi_{resting})\]

Dive information is a mixture model based on behavior (S)

\(\text{Average dive depth}(\psi)\) \[ DiveDepth \sim Normal(dive_{\mu_S},dive_{\tau_S})\]

1.1 Dive profiles per indidivuals

Dive Profiles with Argos timestamps

Dive autocorrelation plots

1.2 Data Statistics before track cut

## # A tibble: 11 x 4
##    Animal     n argos  dive
##     <int> <int> <int> <int>
##  1 131111   548   173   375
##  2 131115  1029   179   850
##  3 131116  1932   457  1475
##  4 131127  9275  2495  6780
##  5 131128   112    52    60
##  6 131130  1165   151  1014
##  7 131132  2292   589  1703
##  8 131133  8496  2077  6419
##  9 131134  2098   653  1445
## 10 131136  6844  1970  4874
## 11 154187  1866   486  1380

## [1] 35657

2 Data Cleaning Rules.

1. All tracks start with an observed argos location.
2. A single track must have one argos location per time step.

2.1 Data after track cut

## # A tibble: 11 x 2
##    Animal     n
##     <int> <int>
##  1 131111   454
##  2 131115  1005
##  3 131116  1892
##  4 131127  8224
##  5 131128    99
##  6 131130   483
##  7 131132  1545
##  8 131133  6873
##  9 131134  1870
## 10 131136  6092
## 11 154187  1836

## # A tibble: 11 x 2
##    Animal Tracks
##     <int>  <int>
##  1 131111      1
##  2 131115      2
##  3 131116      1
##  4 131127     29
##  5 131128      1
##  6 131130      2
##  7 131132      5
##  8 131133     34
##  9 131134      6
## 10 131136     15
## 11 154187      2

sink(“Bayesian/NestedDive.jags”) cat(" model{

pi <- 3.141592653589

#for each if 6 argos class observation error

for(x in 1:6){

##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]

#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
phi[i,g,t,1] <- alpha[state[i,g,t-1]] 
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement in location
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)

}

#Final behavior state
phi[i,g,steps[i,g],1] <- alpha[state[i,g,steps[i,g]-1]] 
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

#first substate
sub_state[i,g,t,1] ~ dcat(sub_lambda[state[i,g,t]])

# loops over observed dive within interval t

for(u in 2:idx[i,g,t]){ 

#Substate, resting or foraging dives?
sub_phi[i,g,t,u,1] <- sub_alpha[state[i,g,t],sub_state[i,g,t,u-1]] 
sub_phi[i,g,t,u,2] <- 1-sub_phi[i,g,t,u,1]
sub_state[i,g,t,u] ~ dcat(sub_phi[i,g,t,u,])
}

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])

#for each dive depth
#dive depth at time t
divedepth[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t],sub_state[i,g,t,u]],depth_tau[state[i,g,t],sub_state[i,g,t,u]])T(0,)

#Assess Model Fit

#Fit dive discrepancy statistics - comment out for memory savings
#eval[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t],sub_state[i,g,t,u]],depth_tau[state[i,g,t],sub_state[i,g,t,u]])T(0,)
#E[i,g,t,u]<-pow((divedepth[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

#dive_new[i,g,t,u] ~ dnorm(depth_mu[state[i,g,t],sub_state[i,g,t,u]],depth_tau[state[i,g,t],sub_state[i,g,t,u]])T(0,)
#Enew[i,g,t,u]<-pow((dive_new[i,g,t,u]-eval[i,g,t,u]),2)/(eval[i,g,t,u])

}
}
}
}

###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016

##Behavioral States
#Movement autocorrelation
gamma[1] ~ dbeta(10,2)  
gamma[2] ~ dbeta(2,10)  

#Transition Intercepts
alpha[1] ~ dbeta(1,1)
alpha[2] ~ dbeta(1,1)

#Probability of init behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

#Probability of init subbehavior switching 
sub_lambda[1] ~ dbeta(1,1)
sub_lambda[2] <- 1 - sub_lambda[1]

#Dive Priors
#Foraging dives
depth_mu[2,1]  ~ dunif(50,250)
depth_sigma[1] ~  dunif(0,90)
depth_tau[2,1] <- 1/pow(depth_sigma[1],2) 

#Resting Dives
depth_mu[2,2] ~ dunif(0,30)
depth_sigma[2] ~ dunif(0,20)
depth_tau[2,2] <- 1/pow(depth_sigma[2],2) 

#Traveling Dives
depth_mu[1,1] ~ dunif(0,100)
depth_sigma[3] ~ dunif(0,20)
depth_tau[1,1] <- 1/pow(depth_sigma[3],2) 

#Dummy traveling substate
depth_mu[1,2]<-0
depth_tau[1,2]<-0.01

#Sub states
#Traveling has no substate
sub_alpha[1,1]<-1 
sub_alpha[1,2]<-0

#ARS has two substates, foraging and resting
#Foraging probability
sub_alpha[2,1] ~ dbeta(1,1)
sub_alpha[2,2] ~ dbeta(1,1)

##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd

#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539

#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156

}"
,fill=TRUE)

sink()

## 
## Processing function input....... 
## 
## Done. 
##  
## Compiling model graph
##    Resolving undeclared variables
##    Allocating nodes
## Graph information:
##    Observed stochastic nodes: 29169
##    Unobserved stochastic nodes: 61143
##    Total graph size: 1632161
## 
## Initializing model
## 
## Adaptive phase..... 
## Adaptive phase complete 
##  
## 
##  Burn-in phase, 9000 iterations x 2 chains 
##  
## 
## Sampling from joint posterior, 1000 iterations x 2 chains 
##  
## 
## Calculating statistics....... 
## 
## Done.

##      user    system   elapsed 
## 33256.967   613.608 35330.266

2.2 Chains

## [1] "model complete"

3 Chains

4 Temporal Variation in Dive Behavior

## # A tibble: 3 x 3
## # Groups:   state [?]
##   state sub_state     n
##   <dbl>     <dbl> <int>
## 1     1         1   236
## 2     2         1   631
## 3     2         2   112

Grouped by stage

5 Behavior Sequence

5.1 Proportion of behaviors

5.2 Simulate dive depths

5.3 Diel

5.4 Month

6 Spatial Prediction

7 Residual Explorations

7.1 Diel

Lines connect individuals

7.2 Monthly

Lines connect individuals

7.3 Posterior Checks

The goodness of fit is a measured as chi-squared. The expected value is compared to the observed value of the actual data. In addition, a replicate dataset is generated from the posterior predicted intensity. Better fitting models will have lower discrepancy values and be Better fitting models are smaller values and closer to the 1:1 line. A perfect model would be 0 discrepancy. This is unrealsitic given the stochasticity in the sampling processes. Rather, its better to focus on relative discrepancy. In addition, a model with 0 discrepancy would likely be seriously overfit and have little to no predictive power.